Admissible Lichnerowicz coordinates for the Schwarzschild metric

TL;DR

This paper extends the construction of harmonic coordinates for the Schwarzschild metric, incorporating additional parameters related to the stress quadrupole moment, and explores similar coordinates in q-harmonic systems.

Contribution

It introduces admissible Lichnerowicz coordinates for the Schwarzschild metric that include a quadrupole moment parameter, expanding previous coordinate systems.

Findings

Global harmonic coordinates are constructed for the Schwarzschild metric with a quadrupole moment.

The supplementary constant relates to the stress quadrupole moment of a singular energy-momentum tensor.

Calculations are extended to q-harmonic coordinates.

Abstract

Global harmonic coordinates for the complete Schwarzschild metric are found for a more general case than that addressed in a previous work by Quan-Hui Liu. The supplementary constant that appears, in addition to the mass, is related to the stress quadrupole moment of a singular energy--momentum tensor. Similar calculations are also carried out in q--harmonic coordinates.